Existence and multiplicity of symmetric solutions for semilinear elliptic equations with singular potentials and critical Hardy–Sobolev exponents

This paper deals with the singular semilinear elliptic problem − div ( | x | − 2 a ∇ u ) = μ u | x | 2 ( 1 + a ) + Q ( x ) | u | p − 2 u | x | b p + σ h ( x , u ) in Ω , u = 0 on ∂ Ω , where Ω ⊂ R N ( N ≥ 3 ) is a smooth bounded domain, 0 ∈ Ω and Ω is G -symmetric with respect to a subgroup G of O ( N ) , 0 ≤ a < N − 2 2 , σ ≥ 0 , 0 ≤ μ < μ ¯ with μ ¯ = ( N − 2 − 2 a 2 ) 2 , a ≤ b < a + 1 , p = p ( a , b ) = 2 N N − 2 ( 1 + a − b ) , Q ( x ) is continuous and G -symmetric on Ω ¯ and h : Ω × R ↦ R is a continuous nonlinearity of lower order satisfying some conditions. Based upon the symmetric criticality principle of Palais and variational methods, we prove several existence and multiplicity results of G -symmetric solutions under certain appropriate hypotheses on σ , Q and h .